Diffie Hellman Problem

Question

I know that in Diffie Hellman, the final key (from Bob's point of view the final key is calculated as follows)

K_B = (g^x mod n)^y mod n, where

x represents Alice's private no.
y represents Bob's private no.
g and n the two public nos.

which can be evaluated as

K = (g^xy mod n) OR (g^yx mod n), where

x represents Alice's private no.
y represents Bob's private no.
g and n the two public nos.

My question is that How does KB = (gx mod n)y mod n evaluates to K = (gxy mod n) OR (gyx mod n). Does the mod operator has a property where (gx mod n)y mod n evaluates to gxy mod n?

Answer 1

I think you have some misconceptions about what $\text{mod}$ means. Let $p$ be an integer, then the "object" $x\text{ mod }p$ is the equivalence class of $x$ in $\mathbb{Z}/p\mathbb{Z}$, that is to say the image of $x$ under the quotient map $$\mathbb{Z}\longrightarrow\mathbb{Z}/p\mathbb{Z},$$ which for simplicity we'll write $[x]$. It is easy to check that indeed $[x]^a = [x^a]$, which is the property you wanted.

Remark: all numbers considered here are integers.